New Horizons in Mathematical Physics

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DOI: 10.22606/nhmp.2019.31001

• Tadayoshi Adachi^1* and Amane Kiyose^{2
  1}Graduate School of Human and Environmental Studies, Kyoto University, Japan ²Department of Mathematics, Graduate School of Science, Kobe University, Japan

In this paper, for N  2, we give a natural derivation of the Avron-Herbst type formula for the time evolution generated by an N-body Hamiltonian with constant electric and magnetic fields. By virtue of the formula, some scattering problems can be reduced to those in the case where the constant electric and magnetic fields are parallel to each other. As an application of the formula, we give the result of the asymptotic completeness for the systems which have the only charged particle and some neutral ones in crossed constant electric and magnetic fields.

Avron-Herbst type formula, constant electric field, constant magnetic field, time evolution.

[1] C. Gérard and I. ?aba, Multiparticle quantum scattering in constant magnetic fields. American Mathematical Society, 2002.

[2] E. Skibsted, “Asymptotic completeness for particles in combined constant electric and magnetic fields, II,” Duke Math. J., vol. 89, no. 2, pp. 307–350, 1997.

[3] T. Adachi and M. Kawamoto, “Avron-Herbst type formula in crossed constant magnetic and time-dependent electric fields,” Lett. Math. Phys., vol. 102, no. 1, pp. 65–90, 2012.

[4] J. Chee, “Landau problem with a general time-dependent electric field,” Ann. Physics, vol. 324, no. 1, pp. 97–105, 2009.

[5] X. P. Wang, “On the magnetic Stark resonances in two-dimensional case,” in Lecture Notes in Phys., 403. Springer, 1992, pp. 211–233.

[6] M. Dimassi and V. Petkov, “Resonances for magnetic Stark Hamiltonians in two-dimensional case,” Int. Math. Res. Not., vol. 2004, no. 77, pp. 4147–4179, 2004.

[7] ——, “Spectral shift function for operators with crossed magnetic and electric fields,” Rev. Math. Phys., vol. 22, no. 4, pp. 355–380, 2010.

[8] ——, “Spectral problems for operators with crossed magnetic and electric fields,” J. Phys. A, vol. 43, no. 47, 474015 (14pp.), 2010.

[9] C. Ferrari and H. Kovarík, “Resonance width in crossed electric and magnetic field,” J. Phys. A, vol. 37, no. 31, pp. 7671–7697, 2004.

[10] ——, “On the exponential decay of magnetic Stark resonances,” Rep. Math. Phys., vol. 56, no. 2, pp. 197–207, 2005.

[11] M. Kawamoto, “Exponential decay property for eigenfunctions of Landau-Stark Hamiltonian,” Rep. Math. Phys., vol. 77, no. 1, pp. 129–140, 2016.

[12] L. M. Lawson and G. Y. H. Avossevou, “Landau problem with time dependent mass in time dependent electric and harmonic background fields,” J. Math. Phys., vol. 59, no. 4, 042109 (9pp.), 2018.

[13] T. Adachi, “Scattering theory for a two-body quantum system in a constant magnetic field,” J. Math. Sci. Univ. Tokyo, vol. 8, no. 2, pp. 243–274, 2001.

[14] ——, “On spectral and scattering theory for N-body Schr?dinger operators in a constant magnetic field,” Rev. Math. Phys., vol. 14, no. 2, pp. 199–240, 2002.

[15] A. Kiyose, “Scattering theory for two-body quantum systems in crossed constant magnetic and electric fields (in Japanese),” Master’s thesis, Kobe University, 2014.

[16] J. Derezinski, “Asymptotic completeness of long-range N-body quantum systems,” Ann. of Math. (2), vol. 138, no. 2, pp. 427–476, 1993.

[17] C. Gérard and I. ?aba, “Scattering theory for N-particle systems in constant magnetic fields, II. Long-range interactions,” Comm. Partial Differential Equations, vol. 20, no. 9-10, pp. 1791–1830, 1995.

[18] Y. Sato, “On two-body scattering in the plane to which crossed constant magnetic and rotating electric fields are impressed (in Japanese),” Master’s thesis, Kobe University, 2017.